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Intuitively, a continuous function is a function whose graph can be drawn without lifting a pen. However, this notion must be made mathematically precise and is denoted in the following manner.

A real function defined on some domain is continuous on a point if the following is satisfied

This definition would have worked except that we run into the problem of having a neighbourhood of . Otherwise, how could one know if there are points sufficiently close to in the domain of ? To make this precise, there are the following two types of definitions.

Continuous functions on an open interval[edit | edit source]

A function is said to be continuous on the open interval if the following is true for each point .